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Multi-point boundary value problem for first order impulsive integro-differential equations with multi-point jump conditions

Abstract

In this article we introduce a new definition of impulsive conditions for boundary value problems of first order impulsive integro-differential equations with multi-point boundary conditions. By using the method of lower and upper solutions in reversed order coupled with the monotone iterative technique, we obtain the extremal solutions of the boundary value problem. An example is also discussed to illustrate our results.

Mathematics Subject Classification 2010: 34B15; 34B37.

1 Introduction

Impulsive differential equations describe processes which have a sudden change of their state at certain moments. Impulse effects are important in many real world applications, such as physics, medicine, biology, control theory, population dynamics, etc. (see, for example [13]). In this article, we consider the following boundary value problem for first order impulsive integro-differential equations (BVP):

x t = f t , x t , F x t , S x t , t J = 0 , T , t t k , Δ x t k = I k l = 1 c k ρ l k x η l k , k = 1 , 2 , m , x 0 + μ k = 1 m l = 1 c k τ l k x η l k = x T ,
(1.1)

where f C(J × R3, R), 0 = t0< t1< t2< · · · < t m < tm+1= T,

F x t = 0 t k t , s x s ds, S x t = 0 T h t , s x s ds,

k C(D, R+), D = {(t, s) J × J: ts}, h C(J × J, R+). I k C(R, R), Δx t k =x t k + -x t k - , t k - 1 < η 1 k < η 2 k << η c k k t k , τ l k , ρ l k 0, l = 1, 2, ..., c k , c k N = {1, 2, ...}, k = 1, 2, ..., m, μ ≥ 0.

The monotone iterative technique coupled with the method of lower and upper solutions is a powerful method used to approximate solutions of several nonlinear problems (see [414]). Boundary value problems for first order impulsive functional differential equations with lower and upper solutions in reversed order have been widely discussed in recent years (see [1520]). However, the discussion of multi-point boundary value problems for first order impulsive functional differential equations is very limited (see [21]). In all articles concerned with applications of the monotone iterative technique to impulsive problems, the authors have assumed that Δx(t k ) = I k (x(t k )), that is a short-term rapid change of the state at impulse point t k depends on the left side of the limit of x(t k ).

Recently, Tariboon [22] and Liu et al. [23] studied some types of impulsive boundary value problems with the impulsive integral conditions

Δx t k = I k t k - τ k t k x s d s - t k - 1 t k - 1 + σ k - 1 x s d s ,k=1,2,,m.
(1.2)

It should be noticed that the terms t k - τ k t k x s ds and t k - 1 t k - 1 + σ k - 1 x s ds of impulsive condition (1.2) illustrate the past memory state on [t k - τ k , t k ] before impulse points t k and the history effects after the past impulse points tk- 1on (tk- 1, tk- 1+ σk- 1], respectively.

The aim of the present article is to discuss the new impulsive multi-point condition

Δx t k = I k l = 1 c k ρ l k x η l k = I k ρ 1 k x η 1 k + + ρ l k x η l k + + ρ c k k x η c k k ,
(1.3)

for t k - 1 < η 1 k < η 2 k < < η c k k t k , k = 1 , 2 , , m . The new jump conditions mean that a sudden change of the state at impulse point t k depends on the multi-point η l k l = 1 , 2 , , c k of past states on (tk- 1, t k ]. We note that if c k = 1, η c k k = t k and ρ c k k =1, then the impulsive condition (1.3) is reduced to the simple impulsive condition Δx(t k ) = I k (x(t k )).

Firstly, we introduce the definitions of lower and upper solutions and formulate some lemmas which are used in our discussion. In the main results, we obtain the existence of extreme solutions for BVP (1.1) by using the method of lower and upper solutions in reversed order and the monotone iterative technique. Finally, we give an example to illustrate the obtained results.

2 Preliminaries

Let J- = J \ {t1, t2, ..., t m }. PC(J, R) = {x: J → R; x(t) is continuous everywhere except for some t k at which x t k - and x t k + exist and x t k - =x t k , k = 1, 2, ..., m}, PC1(J, R) = {x PC(J, R); x'(t) is continuous everywhere except for some t k at which x t k + and x t k - exist and x t k - = x t k }. Let E = PC(J, R) and F=P C 1 J , R , then E and F are Banach spaces with the nomes ||x|| E = suptJ|x(t)| and x F = max x E , x E , respectively. A function xF is called a solution of BVP (1.1) if it satisfies (1.1).

Definition 2.1. A function α 0 F is called a lower solution of BVP (1.1) if:

α 0 t f t , α 0 t , F α 0 t , S α 0 t , t J - , Δ α 0 t k I k l = 1 c k ρ l k α 0 η l k , k = 1 , 2 , , m , α 0 0 + μ k = 1 m l = 1 c k τ l k α 0 η l k α 0 T .

Analogously, a function β 0 F is called an upper solution of BVP (1.1) if:

β 0 t f t , β 0 t , F β 0 t , S β 0 t , t J - , Δ β 0 t k I k l = 1 c k ρ l k β 0 η l k , k = 1 , 2 , , m , β 0 0 + μ k = 1 m l = 1 c k τ l k β 0 η l k β 0 T ,

where t k - 1 < η l k t k , ρ l k , τ l k 0 , l = 1, 2, ..., c k , c k N = {1, 2, ...}, k = 1, 2, ..., m and μ ≥ 0.

Let us consider the following boundary value problem of a linear impulsive integro-differential equation (BVP):

x t - M x t = H F x t + K S x t + v t , t J - , Δ x t k = L k l = 1 c k ρ l k x η l k + I k l = 1 c k ρ l k σ η l k - L k l = 1 c k ρ l k σ η l k , k = 1 , 2 , , m , x 0 + μ k = 1 m l = 1 c k τ l k σ η l k = x T ,
(2.1)

where M > 0, H, K ≥ 0, L k ≥ 0, t k - 1 < η l k t k , τ l k , ρ l k 0, l = 1, 2, ..., c k , c k N = {1, 2, ...}, k = 1, 2,..., m are constants and v(t), σ(t) E.

Lemma 2.1. xFis a solution of (2.1) if and only if x E is a solution of the impulsive integral equation

x t = μ e M t e M T - 1 k = 1 m l = 1 c k τ l k σ η l k - 0 T G t , s P s d s - k = 1 m G t , t k L k l = 1 c k ρ l k x η l k + I k l = 1 c k ρ l k σ η l k - L k l = 1 c k ρ l k σ η l k , t J ,
(2.2)

where P(t) = H(Fx)(t) + K(Sx)(t) + v(t) and

G t , s = e M t - s e M T - 1 , 0 s < t T , e M T + t - s e M T - 1 , 0 t s T .

Proof. Assume that x(t) is a solution of BVP (2.1). By using the variation of parameters formula, we get

x t = x 0 e M t + 0 t e M t - s P s d s + 0 < t k < t e M t - t k L k l = 1 c k ρ l k x η l k + I k l = 1 c k ρ l k σ η l k - L k l = 1 c k ρ l k σ η l k .
(2.3)

Putting t = T in (2.3), we have

x T = x 0 e M T + 0 T e M T - s P s d s + k = 1 m e M T - t k L k l = 1 c k ρ l k x η l k + I k l = 1 c k ρ l k σ η l k - L k l = 1 c k ρ l k σ η l k .
(2.4)

From x 0 +μ k = 1 m l = 1 c k τ l k σ η l k =x T , we obtain

x 0 = - 1 e M T - 1 - μ k = 1 m l = 1 c k τ l k σ η l k + 0 T e M T - s P s d s + k = 1 m e M T - t k L k l = 1 c k ρ l k x η l k + I k l = 1 c k ρ l k σ η l k - L k l = 1 c k ρ l k σ η l k .
(2.5)

Substituting (2.5) into (2.3), we see that x E satisfies (2.2). Hence, x(t) is also the solution of (2.2).

Conversely, we assume that x(t) is a solution of (2.2). By computing directly, we have

G t t , s = M e M t - s e M T - 1 , 0 s < t T , M e M T + t - s e M T - 1 , 0 t s T , = M G t , s .

Differentiating (2.2) for t ≠ t k , we obtain

x t =Mx t +H F x t +K S x t +v t .

It is easy to see that

Δx t k = L k l = 1 c k ρ l k x η l k + I k l = 1 c k ρ l k σ η l k - L k l = 1 c k ρ l k σ η l k .

Since G(0, s) = G(T, s), then x 0 +μ k = 1 m l = 1 c k τ l k σ η l k =x T . This completes the proof.   □

Lemma 2.2. Assume that M > 0, H, K ≥ 0, L k ≥ 0, t k - 1 < η l k t k , ρ l k 0, l = 1, 2, ..., c k , c k N = {1, 2, ...}, k = 1, 2, ..., m, and the following inequality holds:

e M T e M T - 1 0 T H 0 s k s , r d r + K 0 T h s , r d r d s + e M T e M T - 1 k = 1 m L k l = 1 c k ρ l k < 1 .
(2.6)

Then BVP (2.1) has a unique solution.

Proof. For any x E, we define an operator A by

A x t = μ e M t e M T - 1 k = 1 m l = 1 c k τ l k σ η l k - 0 T G t , s H F x s + K S x s + v s d s - k = 1 m G t , t k L k l = 1 c k ρ l k x η l k + I k l = 1 c k ρ l k σ η l k - L k l = 1 c k ρ l k σ η l k , t J ,
(2.7)

where G(t, s) is defined as in Lemma 2.1. Since max t [ 0 , T ] G t , s = e M T e M T - 1 , we have for any x, y E, that

A x - A y E = - 0 T G t , s H 0 s k s , r x r - y r d r + K 0 T h s , r x r - y r d r d s - k = 1 m G t , t k L k l = 1 c k ρ l k x η l k - y η l k e M T e M T - 1 0 T H 0 s k s , r d r + K 0 T h s , r d r d s + e M T e M T - 1 k = 1 m L k l = 1 c k ρ l k x - y E .

From (2.6) and the Banach fixed point theorem, A has a unique fixed point x ¯ E. By Lemma 2.1, x ¯ is also the unique solution of (2.1).   □

Lemma 2.3. Assume thatxFsatisfies

x t M x t + H F x t + K S x t , t J - , Δ x t k L k l = 1 c k ρ l k x η l k , k = 1 , 2 , , m , x 0 x T ,
(2.8)

where M > 0, H, K ≥ 0, L k ≥ 0, t k - 1 < η l k t k , ρ l k 0, l = 1, 2, ..., c k , c k N = {1, 2, ...}, k = 1, 2, ..., m. In addition assume that

e M T 0 T q s d s + k = 1 m L k l = 1 c k ρ l k e - M t k - η l k 1,
(2.9)

whereq t =H 0 t k t , s e - M t - s ds+K 0 T h t , s e - M t - s ds. Then, x(t) ≤ 0 for all t J.

Proof. Set u(t) = x(t)e-Mt for t J , then we have

u t H 0 t k t , s e - M t - s u s d s + K 0 T h t , s e - M t - s u s d s , t J - , Δ u t k L k l = 1 c k ρ l k e - M t k - η l k u η l k , k = 1 , 2 , , m , u 0 e M T u T .
(2.10)

Obviously, the function u(t) and x(t) have the same sign. Suppose, to the contrary, that u(t) > 0 for some t J. Then, there are two cases:

  1. (i)

    There exists a t* J , such that u(t*) > 0 and u(t) ≥ 0 for all t J.

  2. (ii)

    There exists t*, t * J, such that u(t*) > 0 and u(t *) < 0.

Case (i): Equation (2.10) implies that u'(t) ≥ 0 for t J- and Δu(t k ) ≥ 0 for k = 1, 2, ..., m. This means that u(t) is nondecreasing in J. Therefore, u(T) ≥ u(t*) > 0 and u(T) ≥ u(0) ≥ u(T)eMT , which is a contradiction.

Case (ii): Let t* (t i , ti+1], i {0, 1, ..., m}, such that u(t*) = inf {u(t): t J} < 0 and t* (t j , tj+1], j {0, 1, ..., m}, such that u(t*) > 0. We first claim that u(0) ≤ 0. Otherwise, if u(0) > 0, then by (2.10), we have

u t * - u 0 H 0 t * 0 s k s , r e - M s - r u r d r d s + K 0 t * 0 T h s , r e - M s - r u r d r d s + k = 1 i Δ u t k u t * 0 t * q s d s + k = 1 i L k l = 1 c k ρ l k e - M t k - η l k u t * ,
(2.11)

a contradiction, and so u(0) ≤ 0.

If t* < t*, then j ≤ i. Integrating the differential inequality in (2.10) from t* to t*, we obtain

u t * - u t * H t * t * 0 s k s , r e - M s - r u r d r d s + K t * t * 0 T h s , r e - M s - r u r d r d s + k = j + 1 i Δ u t k u t * t * t * q s d s + k = j + 1 i Δ u t k u t * t * t * q s d s + k = j + 1 i L k l = 1 c k ρ l k e - M t k - η l k u η l k u t * 0 T q s d s + k = 1 m L k l = 1 c k ρ l k e - M t k - η l k u t * ,

which is a contradiction to u(t*) > 0.

Now, assume that t*< t*. Since 0 ≥ u(0) ≥ eMTu(T), then u(T) 0. From (2.10), we have

u T - u t * H t * T 0 s k s , r e - M s - r u r d r d s + K t * T 0 T h s , r e - M s - r u r d r d s + k = j + 1 m Δ u t k u t * t * T q s d s + k = j + 1 m L k l = 1 c k ρ l k e - M t k - η l k ,

and u(0) ≥ eMT u(T). In consequence,

u 0 e M T u T e M T u t * +u t * e M T t * T q s d s + k = j + 1 m L k l = 1 c k ρ l k e - M t k - η l k
(2.12)

can be obtained.

If t* = 0, then

u t * e M T u t * + u t * e M T t * T q s d s + k = j + 1 m L k l = 1 c k ρ l k e - M t k - η l k e M T u t * + u t * .

This contradicts the fact that u(t*) > 0.

If t*> 0, we obtain from (2.11),

u t * -u t * 0 t * q s d s + k = 1 i L k l = 1 c k ρ l k e - M t k - η l k u 0 .

This joint to (2.12) yields

u t * - u t * 0 t * q s d s + k = 1 i L k l = 1 c k ρ l k e - M t k - η l k e M T u t * + u t * e M T t * T q s d s + k = j + 1 m L k l = 1 c k ρ l k e - M t k - η l k .

Therefore,

u t * - e M T u t * u t * e M T t * T q s d s + k = j + 1 m L k l = 1 c k ρ l k e - M t k - η l k + u t * 0 t * q s d s + k = 1 i L k l = 1 c k ρ l k e - M t k - η l k u t * e M T t * T q s d s + k = j + 1 m L k l = 1 c k ρ l k e - M t k - η l k + u t * e M T 0 t * q s d s + k = 1 i L k l = 1 c k ρ l k e - M t k - η l k u t * e M T 0 T q s d s + k = 1 m L k l = 1 c k ρ l k e - M t k - η l k u t * .

This is a contradiction and so u(t) ≤ 0 for all t J. The proof is complete.   □

3 Main results

In this section, we are in a position to prove our main results concerning the existence criteria for solutions of BVP (1.1).

For β 0 , α 0 F, we denote

β 0 , α 0 = x F : β 0 t x t α 0 t , t J ,

and we write β0α0 if β0(t) ≤ α0(t) for all t J.

Theorem 3.1. Let the following conditions hold.

(H1) The functions α0and β0are lower and upper solutions of BVP (1.1), respectively, such that β0(t) ≤ α0(t) on J.

(H2) The function f C(J × R3, R) satisfies

f t , x , y , z - f t , x ¯ , y ¯ , z ¯ M x - x ¯ + H y - y ¯ + K z - z ¯ ,

for β 0 t x ¯ t x t α 0 t , F β 0 t y ¯ t y t F α 0 t , S β 0 t z ¯ t z t S α 0 t t J .

(H3) The function I k C(R, R) satisfies

I k l = 1 c k ρ l k x η l k - I k l = 1 c k ρ l k y η l k L k l = 1 c k ρ l k x η l k - y η l k ,

whenever β 0 η l k y η l k x η l k α 0 η l k , l = 1, 2, ..., c k , c k N = {1, 2, ...}, L k ≥ 0, k = 1, 2, ..., m.

(H4) Inequalities (2.6) and (2.9) hold.

Then there exist monotone sequences α n , β n Fsuch that lim n→∞ α n (t) = x*(t), lim n→∞ β n (t) = x* (t) uniformly on J and x*, x*are maximal and minimal solutions of BVP (1.1), respectively, such that

β 0 β 1 β 2 β n x * x x * α n α 2 α 1 α 0 ,

on J, where x is any solution of BVP (1.1) such that β0 (t) ≤ x(t) ≤ α0(t) on J.

Proof. For any σ [β0, α0], we consider BVP (2.1) with

v t =f t , σ t , F σ t , S σ t -Mσ t -H F σ t -K S σ t .

By Lemma 2.2, BVP (2.1) has a unique solution x(t) for t J. We define an operator A by x = , then the operator A is an operator from [β0, α0] to F and A has the following properties.

  1. (i)

    β 0 0, 0α 0;

  2. (ii)

    For any σ l, σ 2 [β 0, α 0], σ lσ 2 implies l 2.

To prove (i), set φ = β0 - β1, where β1 = 0. Then from (Hl) and (2.1) for t J-, we have

φ t = β 0 t - β 1 t , f t , β 0 t , F β 0 t , S β 0 t - M β 1 t + H F β 1 t + K S β 1 t + f t , β 0 t , F β 0 t , S β 0 t - M β 0 t - H F β 0 t - K S β 0 t = M φ t + H F φ t + K S φ t ,
Δ φ t k = Δ β 0 t k - Δ β 1 t k I k l = 1 c k ρ l k β 0 η l k - L k l = 1 c k ρ l k β 1 η l k + I k l = 1 c k ρ l k β 0 η l k - L k l = 1 c k ρ l k β 0 η l k = L k l = 1 c k ρ l k φ η l k , k = 1 , 2 , , m ,

and

φ 0 = β 0 0 - β 1 0 β 0 T - μ k = 1 m l = 1 c k τ l k β 0 η l k - β 1 T + μ k = 1 m l = 1 c k τ l k β 0 η l k = φ T .

By Lemma 2.3, we get that φ(t) ≤ 0 for all t J , i.e., β00. Similarly, we can prove that 0α0.

To prove (ii), let ul = l, u2 = 2, where σlσ2 on J and σl, σ2 [β0, α0]. Set φ = ul - u2. Then for t J- and by (H2), we obtain

φ t = u 1 t - u 2 t = M u 1 t + H F u 1 t + K S u 1 t + f t , σ 1 t , F σ 1 t , S σ 1 t - M σ 1 t - H F σ 1 t - K S σ 1 t - M u 2 t + H F u 2 t + K S u 2 t + f t , σ 2 t , F σ 2 t , S σ 2 t - M σ 2 t - H F σ 2 t - K S σ 2 t M u 1 t - u 2 t + H F u 1 - u 2 t + K S u 1 - u 2 t , = M φ t + H F φ t + K S φ t ,

and by (H 3);

Δ φ t k = Δ u 1 t k - Δ u 2 t k = L k l = 1 c k ρ l k u 1 η l k + I k l = 1 c k ρ l k σ 1 η l k - L k l = 1 c k ρ l k σ 1 η l k - L k l = 1 c k ρ l k u 2 η l k + I k l = 1 c k ρ l k σ 2 η l k - L k l = 1 c k ρ l k σ 2 η l k L k l = 1 c k ρ l k u 1 η l k - u 2 η l k = L k l = 1 c k ρ l k φ η l k , k = 1 , 2 , , m .

It is easy to see that

φ 0 = u 1 0 - u 2 0 = u 1 T - μ k = 1 m l = 1 c k τ l k σ 1 η l k - u 2 T + μ k = 1 m l = 1 c k τ l k σ 2 η l k φ T .

Then by using Lemma 2.3, we have φ(t) ≤ 0, which implies that l2.

Now, we define the sequences {α n }, {β n } such that αn+l= n and βn+l= n . From (i) and (ii) the sequence {α n }, {β n } satisfy the inequality

β 0 β 1 β n α n α 1 α 0 ,

for all n N. Obviously, each α n , β n (n = 1, 2, ...) satisfy

α n t = M α n t + H F α n t + K S α n t + f t , α n - 1 t , F α n - 1 t , S α n - 1 t - M α n - 1 t - H F α n - 1 t - K S α n - 1 t , t J - , Δ α n t k = L k l = 1 c k ρ l k α n η l k + I k l = 1 c k ρ l k α n - 1 η l k - L k l = 1 c k ρ l k α n - 1 η l k , k = 1 , 2 , , m , α n 0 + μ k = 1 m l = 1 c k τ l k α n - 1 η l k = α n T ,

and

β n t = M β n t + H F β n t + K S β n t + f t , β n - 1 t , F β n - 1 t , S β n - 1 t - M β n - 1 t - H F β n - 1 t - K S β n - 1 t , t J - , Δ β n t k = L k l = 1 c k ρ l k β n η l k + I k l = 1 c k ρ l k β n - 1 η l k - L k l = 1 c k ρ l k β n - 1 η l k , k = 1 , 2 , , m , β n 0 + μ k = 1 m l = 1 c k τ l k β n - 1 η l k = β n T .

Therefore, there exist x* and x*, such that lim n→∞ β n = x* and lim n→∞ α n = x* uniformly on J. Clearly, x*, x* are solutions of BVP (1.1).

Finally, we are going to prove that x*, x* are minimal and maximal solutions of BVP (1.1). Assume that x(t) is any solution of BVP (1.1) such that x [β0, α0] and that there exists a positive integer n such that β n (t) ≤ x(t) ≤ α n (t) on J. Let φ = βn+1- x, then for t J-,

φ t = β n + 1 t - x t = M β n + 1 t + H F β n + 1 t + K S β n + 1 t + f t , β n t , F β n t , S β n t - M β n t - H F β n t - K S β n t - f t , x t , F x t , S x t M φ t + H F φ t + K S φ t ,
Δ φ t k = Δ β n + 1 t k - Δ x t k = L k l = 1 c k ρ l k β n + 1 η l k + I k l = 1 c k ρ l k β n η l k - L k l = 1 c k ρ l k β n η l k - I k l = 1 c k ρ l k x η l k L k l = 1 c k ρ l k β n + 1 η l k - x η l k = L k l = 1 c k ρ l k φ η l k , k = 1 , 2 , , m ,

and

φ 0 = β n + 1 0 - x 0 = β n + 1 T - μ k = 1 m l = 1 c k τ l k β n η l k - x T + μ k = 1 m l = 1 c k τ l k x η l k φ T .

Then by using Lemma 2.3, we have φ(t) ≤ 0, which implies that βn+1x on J. Similarly we obtain x ≤ αn+1on J. Since β0x ≤ α0 on J , by induction we get β n ≤ × ≤ α n on J for every n. Therefore, x* (t) ≤ x(t) ≤ x*(t) on J by taking n → ∞. The proof is complete.   □

4 An example

In this section, in order to illustrate our results, we consider an example.

Example 4.1. Consider the BVP

x t = t 3 1 + x t + 1 54 t 0 t t s x s d s 3 + 1 81 t 2 0 1 t s x s d s 3 , t J = 0 , 1 , t 1 2 , Δ x 1 2 = 1 4 1 5 x 1 10 + 3 10 x 1 5 + 1 10 x 3 10 + 1 5 x 2 5 + 1 5 x 1 2 , k = 1 , x 0 + 1 5 1 5 x 1 5 + 2 5 x 3 10 + 2 5 x 1 2 = x 1 ,
(4.1)

where k(t, s) = h(t, s) = ts, m = 1, t 1 = 1 2 , c 1 = 5 , ρ 1 1 = 1 5 , ρ 2 1 = 3 10 , ρ 3 1 = 1 10 , ρ 4 1 = 1 5 , ρ 5 1 = 1 5 , η 1 1 = 1 10 , η 2 1 = 1 5 , η 3 1 = 3 10 , η 4 1 = 2 5 , η 5 1 = 1 2 , τ 1 1 = 0 , τ 2 1 = 1 5 , τ 3 1 = 2 5 , τ 4 1 = 0 , τ 5 1 = 2 5 , μ = 1 5 .

Obviously, α0 = 0, β 0 = - 5 , t 0 , 1 2 - 6 , t 1 2 , 1 are lower and upper solutions for (4.1), respectively, and β0≤ α0.

Let

f t , x , y , z = t 3 1 + x + 1 54 t y 3 + 1 81 t 2 z 3 .

Then,

f t , x , y , z -f t , x ¯ , y ¯ , z ¯ x - x ¯ + 1 2 y - y ¯ + 1 3 z - z ¯ ,

where β 0 t x ¯ t x t α 0 t , F β 0 t y ¯ t y t F α 0 t , S β 0 t z ¯ t z t S α 0 t , t J . It is easy to see that

I 1 l = 1 5 ρ l 1 x η l 1 - I 1 l = 1 5 ρ l 1 y η l 1 = 1 4 l = 1 5 ρ l 1 x η l 1 - y η l 1 ,

whenever β 0 η l 1 y η l 1 x η l 1 α 0 η l 1 , l = 1, ..., 5.

Taking L 1 = 1 4 ,M=1,H= 1 2 ,K= 1 3 , it follows that

e M T 0 T H 0 s k s , r e - M s - r d r + K 0 T h s , r e - M s - r d r d s + k = 1 m L k e - M t k l = 1 c k ρ l k e M η l k = e 0 1 1 2 0 s sr e - s - r d r + 1 3 0 1 sr e - s - r d r d s + 1 4 e - 1 2 1 5 e 1 10 + 3 10 e 1 5 + 1 10 e 3 10 + 1 5 e 2 5 + 1 5 e 1 2 0 . 9287149 1 ,

and

e M T e M T - 1 0 T H 0 s k s , r d r + K 0 T h s , r d r d s + e M T e M T - 1 k = 1 m L k l = 1 c k ρ l k = e e - 1 0 1 1 2 0 s srdr + 1 3 0 1 srdr d s + e e - 1 1 4 1 5 + 3 10 + 1 10 + 1 5 + 1 5 0 . 6261991 < 1 .

Therefore, (4.1) satisfies all conditions of Theorem 3.1. So, BVP (4.1) has minimal and maximal solutions in the segment [β0, α0].

References

  1. Lakshmikantham V, Bainov DD, Simeonov PS: Theory of Impulsive Differential Equations. World Scientific, Singapore 1989.

    Google Scholar 

  2. Bainov DD, Simeonov PS: Impulsive Differential Equations: Periodic Solutions and Applications. Longman Scientific & Technical, New York 1993.

    Google Scholar 

  3. Samoilenko AM, Perestyuk NA: Impulsive Differential Equations. World Scientific, Singapore 1995.

    Google Scholar 

  4. Ding W, Mi J, Han M: Periodic boundary value problems for the first order impulsive functional differential equations. Appl Math Comput 2005, 165: 433-446. 10.1016/j.amc.2004.06.022

    Article  MathSciNet  Google Scholar 

  5. Zhang F, Li M, Yan J: Nonhomogeneous boundary value problem for first-order impulsive differential equations with delay. Comput Math Appl 2006, 51: 927-936. 10.1016/j.camwa.2005.11.028

    Article  MathSciNet  Google Scholar 

  6. Chen L, Sun J: Nonlinear boundary problem of first order impulsive integro-differential equations. J Comput Appl Math 2007, 202: 392-401. 10.1016/j.cam.2005.10.041

    Article  MathSciNet  Google Scholar 

  7. Liang R, Shen J: Periodic boundary value problem for the first order impulsive functional differential equations. J Comput Appl Math 2007, 202: 498-510. 10.1016/j.cam.2006.03.017

    Article  MathSciNet  Google Scholar 

  8. Ding W, Xing Y, Han M: Anti-periodic boundary value problems for first order impulsive functional differential equations. Appl Math Comput 2007, 186: 45-53. 10.1016/j.amc.2006.07.087

    Article  MathSciNet  Google Scholar 

  9. Yang X, Shen J: Nonlinear boundary value problems for first order impulsive functional differential equations. Appl Math Comput 2007, 189: 1943-1952. 10.1016/j.amc.2006.12.085

    Article  MathSciNet  Google Scholar 

  10. Luo Z, Jing Z: Periodic boundary value problem for first-order impulsive functional differential equations. Comput Math Appl 2008, 55: 2094-2107. 10.1016/j.camwa.2007.08.036

    Article  MathSciNet  Google Scholar 

  11. Wang X, Zhang J: Impulsive anti-periodic boundary value problem of first-order integro-differential equations. J Comput Appl Math 2010, 234: 3261-3267. 10.1016/j.cam.2010.04.024

    Article  MathSciNet  Google Scholar 

  12. Song G, Zhao Y, Sun X: Integral boundary value problems for first order impulsive integro-differential equations of mixed type. J Comput Appl Math 2011, 235: 2928-2935. 10.1016/j.cam.2010.12.007

    Article  MathSciNet  Google Scholar 

  13. Nieto JJ, Rodríguez-López R: Existence and approximation of solutions for nonlinear functional differential equations with periodic boundary value conditions. Comput Appl Math 2000, 40: 433-442. 10.1016/S0898-1221(00)00171-1

    Article  Google Scholar 

  14. Nieto JJ, Rodríguez-López R: Periodic boundary value problem for non-Lipschitzian impulsive functional differential equations. J Math Anal Appl 2006, 318: 593-610. 10.1016/j.jmaa.2005.06.014

    Article  MathSciNet  Google Scholar 

  15. He Z, Yu J: Periodic boundary value problem for first-order impulsive functional differential equations. J Comput Appl Math 2002, 138: 205-217. 10.1016/S0377-0427(01)00381-8

    Article  MathSciNet  Google Scholar 

  16. He Z, Yu J: Periodic boundary value problem for first-order impulsive ordinary differential equations. J Math Anal Appl 2002, 272: 67-78. 10.1016/S0022-247X(02)00133-6

    Article  MathSciNet  Google Scholar 

  17. Chen L, Sun J: Nonlinear boundary value problem of first order impulsive functional differential equations. J Math Anal Appl 2006, 318: 726-741. 10.1016/j.jmaa.2005.08.012

    Article  MathSciNet  Google Scholar 

  18. Chen L, Sun J: Nonlinear boundary value problem for first order impulsive integro-differential equations of mixed type. J Math Anal Appl 2007, 325: 830-842. 10.1016/j.jmaa.2006.01.084

    Article  MathSciNet  Google Scholar 

  19. Wang G, Zhang L, Song G: Extremal solutions for the first order impulsive functional differential equations with upper and lower solutions in reversed order. J Comput Appl Math 2010, 235: 325-333. 10.1016/j.cam.2010.06.014

    Article  MathSciNet  Google Scholar 

  20. Zhang L: Boundary value problem for first order impulsive functional integro-differential equations. J Comput Appl Math 2011, 235: 2442-2450. 10.1016/j.cam.2010.10.045

    Article  MathSciNet  Google Scholar 

  21. Zhang Y, Zhang F: Multi-point boundary value problem of first order impulsive functional differential equations. J Appl Math Comput 2009, 31: 267-278. 10.1007/s12190-008-0209-2

    Article  MathSciNet  Google Scholar 

  22. Tariboon J: Boundary value problems for first order functional differential equations with impulsive integral conditions. J Comput Appl Math 2010, 234: 2411-2419. 10.1016/j.cam.2010.03.007

    Article  MathSciNet  Google Scholar 

  23. Liu Z, Han J, Fang L: Integral boundary value problems for first order integro-differential equations with impulsive integral conditions. Comput Math Appl 2011, 61: 3035-3043. 10.1016/j.camwa.2011.03.094

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This research is supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand.

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Thaiprayoon, C., Samana, D. & Tariboon, J. Multi-point boundary value problem for first order impulsive integro-differential equations with multi-point jump conditions. Bound Value Probl 2012, 38 (2012). https://doi.org/10.1186/1687-2770-2012-38

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